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Properties

Label 20.12.1457098065...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 419^{4}\cdot 695771^{2}$
Root discriminant $28.72$
Ramified primes $5, 419, 695771$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1040

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -142, 457, -944, 1030, 1013, -4084, 2874, 596, -478, 195, -1598, 1209, -62, -21, -57, -50, 49, -3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 3*x^18 + 49*x^17 - 50*x^16 - 57*x^15 - 21*x^14 - 62*x^13 + 1209*x^12 - 1598*x^11 + 195*x^10 - 478*x^9 + 596*x^8 + 2874*x^7 - 4084*x^6 + 1013*x^5 + 1030*x^4 - 944*x^3 + 457*x^2 - 142*x + 19)
 
gp: K = bnfinit(x^20 - 5*x^19 - 3*x^18 + 49*x^17 - 50*x^16 - 57*x^15 - 21*x^14 - 62*x^13 + 1209*x^12 - 1598*x^11 + 195*x^10 - 478*x^9 + 596*x^8 + 2874*x^7 - 4084*x^6 + 1013*x^5 + 1030*x^4 - 944*x^3 + 457*x^2 - 142*x + 19, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 3 x^{18} + 49 x^{17} - 50 x^{16} - 57 x^{15} - 21 x^{14} - 62 x^{13} + 1209 x^{12} - 1598 x^{11} + 195 x^{10} - 478 x^{9} + 596 x^{8} + 2874 x^{7} - 4084 x^{6} + 1013 x^{5} + 1030 x^{4} - 944 x^{3} + 457 x^{2} - 142 x + 19 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(145709806576045624081650390625=5^{10}\cdot 419^{4}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.72$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 419, 695771$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{101} a^{18} + \frac{3}{101} a^{17} - \frac{47}{101} a^{16} - \frac{26}{101} a^{15} + \frac{9}{101} a^{14} - \frac{35}{101} a^{13} - \frac{4}{101} a^{12} - \frac{37}{101} a^{11} - \frac{27}{101} a^{10} - \frac{5}{101} a^{9} - \frac{29}{101} a^{8} + \frac{34}{101} a^{7} + \frac{12}{101} a^{6} - \frac{49}{101} a^{5} - \frac{40}{101} a^{4} - \frac{15}{101} a^{3} - \frac{6}{101} a^{2} + \frac{28}{101} a - \frac{22}{101}$, $\frac{1}{5967257313241109831180971} a^{19} + \frac{15004332563911713268745}{5967257313241109831180971} a^{18} - \frac{989516071054853896986090}{5967257313241109831180971} a^{17} + \frac{2525142358898279388690867}{5967257313241109831180971} a^{16} - \frac{2774214587209840715695679}{5967257313241109831180971} a^{15} - \frac{1111727360843417551738423}{5967257313241109831180971} a^{14} - \frac{1416296096099741152695072}{5967257313241109831180971} a^{13} - \frac{1649808461703525495654995}{5967257313241109831180971} a^{12} + \frac{1378823899503995941695043}{5967257313241109831180971} a^{11} - \frac{738185579979933818787683}{5967257313241109831180971} a^{10} - \frac{1091876255949761581455681}{5967257313241109831180971} a^{9} - \frac{66449300701322603049520}{5967257313241109831180971} a^{8} + \frac{724388381049927410956688}{5967257313241109831180971} a^{7} - \frac{2088847453747764937267856}{5967257313241109831180971} a^{6} - \frac{388669476427429459400710}{5967257313241109831180971} a^{5} + \frac{89022783214549253953324}{5967257313241109831180971} a^{4} + \frac{844610633903766206812329}{5967257313241109831180971} a^{3} + \frac{349660559251286051635362}{5967257313241109831180971} a^{2} + \frac{498258288688822821296965}{5967257313241109831180971} a + \frac{1051077461821442579530150}{5967257313241109831180971}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 19270931.5245 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1040:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1040 are not computed
Character table for t20n1040 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
419Data not computed
695771Data not computed